Understanding flat substructures of non-positively curved spaces reveals strong geometric, topological, and group theoretic information. I will present an extension of Benoist' 2006 result on triangles in 3-dimensional divisible real convex projective domains to a statement about codimension-1 simplices in every dimension greater than two. We recover Benoist' `geometric JSJ-decomposition', and a higher dimensional analogue for convex projective manifolds. Speaker(s): Martin Bobb (U Texas)